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Exercises 1.3 Exercises

1.

Let \(A = \{1-1/n : n\in\N\}\text{.}\) Find the set of upper bounds of \(A\) and the set of lower bounds of \(A\text{.}\) Is \(A\) bounded?

2.

Give an example of bounded sets \(A_n\subseteq\R\) (\(n\in\N\)) such that \(\bigcup_{n=1}^{\infty}A_n\) is not bounded. Give an example of two non-empty sets bounded above whose intersection is empty.

3.

For each set, decide whether it is an interval. Justify your answer.
  • \(\displaystyle \Q\cap(0,1)\)
  • \(\displaystyle [0,1)\cup\{2\}\)
  • \(\displaystyle \{x\in\R : x^2\lt 2\}\)
  • \(\displaystyle \{x\in\R : x\le 3\}\)

4.

Suppose \(A\) is a nonempty subset of a bounded set \(B\subseteq\R\text{.}\) Show that
\begin{equation*} \inf B \le \inf A \le \sup A \le \sup B. \end{equation*}

5.

Compute the intersection of the two intervals \(I = (-2,5]\) and \(J=[1,\infty)\text{,}\) and express your answer in interval notation.

6.

Give one example of two intervals whose union is an interval, and one example of two intervals whose union is not an interval.

7.

Let \(a\lt b\text{.}\) Define \(f:(a,b)\to(-1,1)\) by \(f(x)=\dfrac{x-c}{r}\text{,}\) where \(c=(a+b)/2\) and \(r=(b-a)/2\text{.}\) Show that \(f\) is a bijection.

8.

Is each of the following sets finite, countably infinite, or uncountable?
  • \(\displaystyle \N\)
  • \(\displaystyle (0,1)\cap\Q\)
  • \(\displaystyle (0,1)\)
  • \(\displaystyle \R\setminus\Q\)

9.

Prove that the least upper bound property is equivalent to: every non-empty subset of \(\R\) that is bounded below has an infimum.

10.

Prove that \(\N\) is not bounded above in \(\R\text{.}\)

11.

Let \(A=\{q\in\Q:q>0,\ q^2<2\}\text{.}\) Show that \(A\) is non-empty and bounded above in \(\Q\text{,}\) but has no supremum in \(\Q\text{.}\)
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